Fourier transform integral for even function proof

Question

Using fourier transform prove that: If $f(t)$ is even $$F(v)= 2\int_0^\infty f(t)cos(2\pi\nu t) \,\mathrm dt$$

How to do that?

Answer 1

\begin{align} \mathcal F\{f(t)\}=F(\nu)&=\int_{-\infty}^{\infty}f(t)\mathrm e^{-i 2\pi\nu t}\mathrm dt\\ &=\int_{-\infty}^{0}f(t)\mathrm e^{-i 2\pi\nu t}\mathrm dt+\int_{0}^{\infty}f(t)\mathrm e^{-i 2\pi\nu t}\mathrm dt\\ \quad&=\int_{0}^{\infty}f(-t)\mathrm e^{i 2\pi\nu t}\mathrm dt+\int_{0}^{\infty}f(t)\mathrm e^{-i 2\pi\nu t}\mathrm dt\\ &=\int_{0}^{\infty}f(t)\underbrace{\left(\mathrm e^{i 2\pi\nu t}+\mathrm e^{-i 2\pi\nu t}\right)}_{2\cos(2\pi\nu t)}\mathrm dt\\ &=2\int_{0}^{\infty} f(t)\cos(2\pi\nu t)\mathrm dt \end{align}